Angle Between Two Lines

Let y = m1x + c1 and y = m2x + c2 be the equations of two lines in a plane where,

m1 = slope of line 1
c1 = y-intercept made by line 1 ​                                                                                                                                                

m2 = slope of line 2
c2 = y-intercept made by line 2

Angle between two lines

<BAX = θ1
<DCX = θ2

∴ m1 = tan θ1 and 
  m2 = tan θ2                                        
Let the angle between the lines AB and CD be Ø (<APC) then,
Formula for angle between two lines

Also if we consider <APD as the angle between lines,

Formula for angle between two lines  

Hence, the angle between two lines is, 

                                                 

Condition for perpendicularity

The two lines are perpendicular means. Ø = 90°
Condition for two lines to be perpendicular to each other

Thus, the lines are perpendicular if the product of their slope is -1.

Condition for parallelism

The two lines are perpendicular means, Ø = 0°

   Condition for two lines to be parallel to each other               

Thus, the lines are parallel if their slopes are equal.

Angle Between Two Lines Examples

1. Find the angle between the lines 2x-3y+7 = 0 and 7x+4y-9 = 0.

Solution:

Comparing the equation with equation of straight line, y = mx + c,

Slope of line 2x-3y+7=0 is (m1) = 2/3

Slope of line 7x+4y-9=0 is (m2) = -7/4

Let, Ø be the angle between two lines, then

  Example of calculation of examples between two lines using slope

2. Find the equation of line through point (3,2) and making angle 45° with the line x-2y = 3.

Solution:

Let m be the slope of the required line passing through (3,2). So, using slope point form, its equation is

y-2 = m(x-3)      ---------   (i)

Slope of line x-2y = 3 is 1/2.

Since, these lines make an angle of 45° so,

                        Straight line example

Substituting values of m in equation (i), we get

y-2 = 3(x-3) and y-2 = -(x-3)/3

or, 3x-y-7 = 0 and x+3y-9 = 0 are the required equations of line.